In fact, it is also compatible with Poincaré to regard the translation generators as the modes of a primary operator in which the left, right, or both conformal weights are \(-\frac{1}{2}\)  as opposed to \(\frac{3}{2}\) . Such primary operators can be constructed by taking various light or shadow transforms of the \(\Delta = 1\)  conformally soft graviton. However, these modes do not automatically form a closed algebra with the global conformal generators and require an additional assumption of a truncated mode expansion as presented, for example, in [1]}, [2]}. More details on this alternative mode expansion are provided in Appendix REF .
